Parabolic Patterns

The illustration shows the graphs of fifteen functions. Two of them have equations y=x^2 and y=-(x-4)^2. Find the equations of all the other graphs.

More Parabolic Patterns

The illustration shows the graphs of twelve functions. Three of them have equations y=x^2, x=y^2 and x=-y^2+2. Find the equations of all the other graphs.

Parabolas Again

Here is a pattern composed of the graphs of 14 parabolas. Can you find their equations?

Lattice Points

Stage: 5 Challenge Level:

(1) Show that if there is one lattice point (point with integer coordinates) on the parabola $$y=ax^2$$ then there are infinitely many.

(2) Find all the lattice points on the hyperbola $$x^2 - y^2 = 84.$$